Question

The displacement of a string is given by $y(x,t)=0.06\sin (2\pi x/3)\cos (120\pi t)$ where $x$ and $y$ are in $m$ and $t$ in $s$. The length of the string is $1.5m$ and its mass is $3.0\times 10^{-2}kg$.

• A. It represents a progressive wave of frequency 60 Hz
• B. It represents a stationary wave of frequency 60 Hz
• C. It is the result of superposition of two waves of wavelength 3 m, frequency 60 Hz each travelling with a speed of 180 m / s in opposite direction
• D. Amplitude of this wave is constant

Answer 1

Answer: (B), (C)

(a) The given equation is $y(x,t)=0.06\sin \left(\frac{2\pi x}{3}\right)\cos (120\pi t)$
(b) As terms involving $x$ and $t$ are independent of each other, the given equation represents a stationary wave.
(c) Compare the given equation with the standard form of equation of stationary wave
$y(x,t)=2r\sin kx\cos \omega t$
$k=\frac{2\pi }{\lambda }=\frac{2\pi }{3}$
$\therefore \lambda =3m$
$\omega =120\pi$
$\therefore v=\frac{\omega }{2\pi }=\frac{120\pi }{2\pi }=60Hz$
and $v=v\lambda =60\times 3=180m/s$
Hence the given stationary wave is the result of superposition of two waves of wavelength $3m$ and frequency $60Hz$ each, travelling with a velocity of $180m/s$ in opposite directions.